Path: utzoo!utgpu!watserv1!watmath!mks.com!tslwat!louk
From: louk@tslwat.UUCP (Lou Kates)
Newsgroups: comp.lang.apl
Subject: Re: Statistical Functions in J
Message-ID: <356@tslwat.UUCP>
Date: 19 May 91 15:29:42 GMT
References: <1991May12.145907.19563@yrloc.ipsa.reuter.COM>
Reply-To: louk@tslwat.UUCP (Lou Kates)
Organization: Teleride Sage, Ltd., Waterloo
Lines: 36

In article <1991May12.145907.19563@yrloc.ipsa.reuter.COM> hui@yrloc.ipsa.reuter.COM (Roger Hui) writes:
>The definitions of mean (m.), normalize (n.), and spread (s.) have
>changed in response to a detailed critique by Professor Fraser Jackson
>of the Victoria University of Wellington (uunet!matai.vuw.ac.nz!jackson).
>Professor Jackson's comments are as follows (minor editing by me):
> 
>--------------------------------------------------------------------
> 
>The concept of expectation is central to both probability and
>statistics.  Indeed it is quite possible to axiomatise
>probability theory in terms of expectations rather than
>probabilities.  One didactic advantage of doing so is that the
>whole apparatus of measure theory essential for an
>axiomatisation based on probabilities can be introduced much
>later.  

Nevertheless, I believe the key  concept here is not expectation,
probability or  measure  but regression and projection. From this
viewpoint the old APL's domino  operator (or regression operator)
had it correct and the above suggestions are a step backwards.

The mean    of  a  vector  V   is the  regression coefficient  of
projecting  V onto a vector  of all ones. The space orthogonal to
this vector of ones  is the deviation space and the length of the
projection of  V onto  this  deviation  space   divided   by  the
dimensionality of the deviation  space (which is  the length of V
minus one) is the standard deviation.

This fits    in  with  the   geometric  interpretation of  linear
statistical  methods that is commonly taught to statisticians and
is  the standard  way   of visualizing and unifying a variety  of
statistical techniques including regression, analysis of variance
and time series analysis.

Lou Kates, Teleride Sage Ltd., louk%tslwat@watmath.waterloo.edu
519-725-0646